arXiv subjects
Publications and source records attributed to Oishee Banerjee.
We prove that the space of algebraic maps between two smooth projective varieties, under certain conditions, admit a configuration space model, thereby obtaining an algebro-geometric analogue of Bendersky-Gitler's result on topological function spaces. Our result is a natural higher dimensional counterpart of \cite[Theorem 3]{Ban24}.
We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(Σ)$, denoted by the rational polyhedral fan $Σ$, stabilizes. As an arithmetic consequence we obtain a resolution of the Batyrev-Manin conjecture for toric varieties over global function fields in all but finitely many characteristics.
We compute the (stable) étale cohomology of $\mathrm{Hom}_{n}(C, \mathcal{P}(\vecλ))$, the moduli stack of degree $n$ morphisms from a smooth projective curve $C$ to the weighted projective stack $\mathcal{P}(\vecλ)$, the latter being a stacky quotient defined by $\mathcal{P}(\vecλ) := \left[\mathbb{A}^N-\{0\}/\mathbb{G}_m\right]$, where $\mathbb{G}_m$ acts by weights $\vecλ = (λ_0, \cdots, λ_N) \in \mathbb{Z}^N_{+}$. Our key ingredient is formulating and proving the étale cohomological descent over the category $ΔS$, the symmetric (semi)simplicial category. An immediate arithmetic consequence is the resolution of the geometric Batyrev--Manin type conjecture for weighted projective stacks over global function fields. Along the way, we also analyze the intersection theory on weighted projectivizations of vector bundles on smooth Deligne-Mumford stacks.
In \cite{Vakil13} Vakil and Wood made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of $\Sym^n(¶^1)$. In this note, we disprove one of them thereby obtaining a counterexample to the principle of Occcam's razor for Hodge structures; and we prove that the other conjecture, with a minor correction, holds true.
In the simplicial theory of hypercoverings, we replace the indexing category $\Delta$ by the \emph{symmetric simplicial category} $\Delta S$ and study (a class of) $\Delta S$-hypercoverings, which we call \emph{spaces admitting symmetric (semi)simplicial filtration}. For $\Delta S$-hypercoverings we construct a spectral sequence, somewhat like the \v{C}ech-to-derived category spectral sequence. The advantage of working on $\Delta S$ is that all of the combinatorial complexities that come with working on $\Delta$ are bypassed, giving simpler, unified proof of known results like the computation of (in some cases, stable) singular cohomology (with $\mathbb{Q}$ coefficients) and et al e cohomology (with $\mathbb{Q}_{\ell}$ coefficients) of the moduli space of degree $n$ maps $C\to \mathbb{P}^r$, $C$ a smooth projective curve of genus $g$, of unordered configuration spaces etc. as well as new: that of the moduli space of smooth sections of a fixed $\mathfrak{g}^r_d$ that is $m$-very ample for some $m$.
In this paper we study the moduli spaces $Simp^m_n$ of degree $n+1$ morphisms $ \mathbb{A}^1_{K} \to \mathbb{A}^1_{K}$ with "ramification length $ n+1$. As a by-product we obtain that $H^*(Simp^m_n(\mathbb{C}); \mathbb{Q})$ is independent of $n$, thus implying rational cohomological stability. When $char K>0$ our methods compute $H^*_{et}(Simp^m_n; \mathbb{Q}_{\ell})$ provided $char K>n+1$ and show that the étale cohomology groups in positive characteristics do not stabilize.